nLab on horizontal categorification. Example: L-infinity algebras, I think.
http://mathoverflow.net/questions/4841/what-precisely-is-categorification
See also n-Lab on categorification
arXiv:1011.0144 Lectures on algebraic categorification from arXiv Front: math.CT by Volodymyr Mazorchuk This is a write-up of the lectures given by the author during the Master Class “Categorification” at Århus University, Denmark in October 2010.
Email from Alexander Palen Ellis: My advisor’s (and by extension, I guess, my) main project right now is categorification of representations. The basic idea given an algebra A, we want to find a category and some distinguished endo-functors acting on something, such that when you take K_0 of the whole mess you recover your original algebra A acting on various modules for that algebra. The case we care about most is when A is a quantum group, say U_q(g) – if this can be successfully done, then the Hopf algebra structure on U_q(g) lifts to a “Hopf category” structure on the categorification. This in turn, by work of Frenkel, should yield interesting 4D TQFTS (there are already 3D ones from U_q(g) itself). More generally, one expects categorification to “jack up the dimension” of existing invariants of things, yielding much richer structures. In particular, you often get integrality / positivity results (since you can interpret certain structure constants in an algebra as “the number of times you’ve applied a certain functor”).
For categorification of the character of a linear rep, see Toen preprint: infty-categories monoidales rigides etc. File Toen web prepr dag-loop.pdf.
Toen: Note of Chern character, loop spaces and derived algebraic geometry. File Toen web publ Abel-2007.pdf. Contains quite a lot of material of categorification.
Toen: Note of Chern character, loop spaces and derived algebraic geometry. File Toen web publ Abel-2007.pdf. Notion of derived cat sheaves, a categorification of the notion of complexes of sheaves of O-modules on schemes (also quasi-coh and perfect versions). Chern character for these categorical sheaves, a categorified version of the Chern char for perfect complexes with values in cyclic homology. Idea of categorical sheaves: For X a scheme, should have a symmetric monoidal 2-cat Cat(X) which is a categorification of Mod(X), in the sense that Mod(X) should be the cat of endomorphisms of the unit objects in Cat(X). More details. Categorification of homological algebra and dg-cats. See Derived categorical sheaves for a longer summary of this paper.
“One common feature of recent trends is “categorification”, often synonymous with “geometrization”. Categorification stands for the passage from a traditional mathematical object to its higher categorical analogue, and, more loosely, for the emphasis on categories instead of particular objects. The categories involved are typically of geometric nature (categories of sheaves of certain kind) and are constructed in a homological framework, i.e., they are triangulated categories, or refinements of these. Examples in representation theory include geometric Langlands duality, …, homological mirror symmetry and, more generally, focus on derived categories of coherent sheaves in algebraic geometry, which is a categorification of standard cohomology theories.”
nLab page on Categorification